I'm struggling to reflect curves over the line of $ y = x $, I have no trouble reflecting shapes over the x or y axis, but always struggle when it comes to $ y = x$. I always have to resort to plotting which takes up a lot of time.
Could someone give me any advice on how to reflect curves over the line $y=x$
Hint: Reflections about the line $y = x$ is accomplished by interchanging the $x$ and the $y$ coordinates. Thus for $y = f(x)$, the reflection about the line $y = x$ is accomplished by $x = f(y)$.
For example, the reflection about the line $y = x$ for $y = x^2$ is the equation $x = y^2$. Hope it helps.
If the curve is complicated, you can just take particular, interesting points and switch the coordinates. Every $x$-intercept becomes a $y$ intercept and vv., for instance. Max's, min's, cusps, asymptotes and other features can be similarly transposed. Then connect the dots appropriately.